A q-analog of Ljunggren's binomial congruence

A q-analog of Ljunggren's binomial congruence
Armin Straub — Discrete Mathematics and Theoretical Computer Science — Special volume for FPSAC 2011 — 2011, Pages 897-902


We prove a \(q\)-analog of a classical binomial congruence due to Ljunggren which states that \[ \binom{a p}{b p} \equiv \binom{a}{b} \] modulo \(p^3\) for primes \(p\ge5\). This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing \(q\)-analogs. Our congruence generalizes an earlier result of Clark.


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    author = {Armin Straub},
    title = {A $q$-analog of {L}junggren's binomial congruence},
    journal = {Discrete Mathematics and Theoretical Computer Science},
    year = {2011},
    pages = {897--902},